[seqfan] Re: Sequences with Jacobi symbols in definitions

[Tomasz Ordowski]

PS. It seems that A = {3, 5, 7, 11, 13},
so P_a = 3*5*7*11*13 = 13#/2 = 15015.
is the period of the first sequence (a_n).

It also seems that B  = {3, 5, 7, 9, 11},
so P_b = LCM(B) = 5*7*9*11 = 11!!/3 = 3465
is the period of the fourth sequence (b_n).

Tom Ordo


pon., 23 paź 2023 o 18:51 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> ... and the issue of their periodicity.
>
> Dear number lovers!
>
> Let a(n) be the smallest odd prime p such that n^{(p+1)/2} == n (mod p).
> For n > 0 : 3, 7, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 13, 3, 3, 5,
> 3, 3, 7, ...
> Note that a(n) is the smallest odd prime p for which (n / p) >= 0.
> If this sequence is bounded, a(n) <= q, then it is periodic with
> the period P = LCM(A) <= q#/2; see b(n) at the end.
>
> Let A(n) be the smallest odd prime p such that n^{(p-1)/2} == 1 (mod p).
> For n > 0 : 3, 7, 11, 3, 11, 5, 3, 7, 5, 3, 5, 11, 3, 5, 7, 3, 13, 7, 3,
> 11, 5, ...
> A(n) is the smallest odd prime p for which (n / p) = 1.
> This sequence is unbounded: A(p#/2) > p.
> Of course A(n) >= a(n).
>
> Let B(n) be the smallest odd k > 1 such that (n / k) = 1.
> For n > 0 : 3, 7, 11, 3, 9, 5, 3, 7, 5, 3, 5, 11, 3, 5, 7, ...
> I did not find this basic sequence in the OEIS.
> This sequence is also unbounded.
>
> Let b(n) be the smallest odd k > 1 such that n^{(k+1)/2} == n (mod k).
> b(n) : 3, 7, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 9, 3, 3, 5, 3, 3,
> 7, ...
> If b(n) = m, then (n / m) >= 0. It seems that b(n) <= A326614(n).
> If this sequence is bounded, then it is periodic with the period
> P = LCM(B), where B is the set of all pairwise distinct terms.
> Note that n^{(1729+1)/2} == n (mod 1729) for every n > 0,
> where 1729 is the smallest absolute Euler pseudoprime;
> A033181 (see third comment): https://oeis.org/A033181
> Thus b(n) <= 1729 (perhaps much smaller).
> So, as said, this sequence is periodic.
> How long is this period?
>
> Similarly, such a sequence, but of composite numbers,
> (9, 341, 121, 341, 65, 15, 21, 9, 9, 9, 33, 33, 21, ...)
> is bounded, but has a very very long period,
> longer than the Euler primary pretenders;
> A309316: https://oeis.org/A309316
>
> Best,
>
> Tom Ordo
> ___________
> See A326614: https://oeis.org/A326614
> Smallest Euler-Jacobi pseudoprime to base n.
> __________
> Cf. A354689: https://oeis.org/A354689
> Smallest Euler pseudoprime to base n.
>
>